A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

  1. Argimiro Arratia 1
  2. Alejandra Cabaña 2
  3. Enrique M. Cabaña 3
  1. 1 Universitat Politècnica de Catalunya
    info

    Universitat Politècnica de Catalunya

    Barcelona, España

    ROR https://ror.org/03mb6wj31

  2. 2 Universitat Autònoma de Barcelona
    info

    Universitat Autònoma de Barcelona

    Barcelona, España

    ROR https://ror.org/052g8jq94

  3. 3 Universidad de la República
    info

    Universidad de la República

    Montevideo, Uruguay

    ROR https://ror.org/030bbe882

Montrer des affiliations +
Revue:
Sort: Statistics and Operations Research Transactions

ISSN: 1696-2281

Année de publication: 2016

Volumen: 40

Número: 2

Pages: 267-302

Type: Article

DIALNET GOOGLE SCHOLAR lock_openAccès ouvert editor

D'autres publications dans: Sort: Statistics and Operations Research Transactions

Résumé

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same Lévy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p−1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.

€ Voir financement

Information sur le financement

Supported by Spain's MINECO project APCOM (TIN2014-57226-P) and Generalitat de Catalunya 2014SGR 890 (MACDA). Supported by Spain's MINECO project MTM2015-69493-R.

Financeurs

Références bibliographiques

  • Barndorff-Nielsen, O.E. (2001). Superposition of Ornstein-Uhlenbeck type processes. Theory of Probability and Its Applications, 45, 175–194.
  • Bergstrom, A.R. (1984). Continuous time stochastic models and issues of aggregation over time.Handbook of Econometrics, II, 1145–1212.
  • Bergstrom, A.R. (1996). Survey of continuous-time econometrics. In Dynamic Disequilibrium Modeling: Theory and Applications: Proceedings of the Ninth International Symposium in Economic Theory and Econometrics, volume 9, page 1. Cambridge University Press.
  • Box, G.E.P. Jenkins, G.M. and Reinsel, G.C. (1994). Time Series Analysis, Forecasting and Control. Prentice Hall.
  • Brockwell, P.J. (2004). Representations of continuous-time ARMA processes. Journal of Applied Probability, 41, 375–382.
  • Brockwell, P.J. (2009). Lévy–driven continuous–time ARMA processes. In Handbook of Financial Time Series, pages 457–480. Springer.
  • Cleveland, W.S. (1971). The inverse autocorrelations of a time series and their applications. Technometrics, 14, 277–298.
  • Doob, J.L. (1944). The elementary Gaussian processes. Annals of Mathematical Statistics, 15, 229–282.
  • Durbin, J. (1961). Efficient fitting of linear models for continuous stationary time-series from discrete data. Bulletin of the International Statistical Institute, 38, 273–282.
  • Durbin, J. and Koopman, S.J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.
  • Eliazar, I. and Klafter, J. (2009). From Ornstein-Uhlenbeck dynamics to long-memory processes and Fractional Brownian motion. Physical Review E, 79, 021115.
  • Granger, C.W.J. (1980). Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, 14, 227–238.
  • Granger, C.W.J. and Morris, M.J. (1976). Time series modelling and interpretation. Journal of the Royal Statistical Society. Series A, 139, 246–257.
  • Jongbloed, G., van der Meulen, F.H. and van der Vaart, A.W. (2005). Nonparametric inference for Lévydriven Ornstein-Uhlenbeck processes. Bernoulli, 11, 759–791.
  • Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer Science & Business Media.
  • McLeod, A.I. and Zhang, Y. (2006). Partial autocorrelation parameterization for subset autoregression. Journal of Time Series Analysis, 27, 599–612.
  • Nieto, B., Orbe, S. and Zarraga, A. (2014). Time-Varying Market Beta: Does the estimation methodology matter? SORT, 31, 13–42.
  • R Core Team. (2015). R: A Language and Environment for Statistical Computing. Technical report, R Foundation for Statistical Computing, Vienna, Austria.
  • Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distribution, volume 68 of Cambridge Studies in Advance Mathematics. Cambridge University Press.
  • Thornton, M.A. and Chambers, M.J. (2013). Continuous-time autoregressive moving average processes in discrete time: representation and embeddability. Journal of Time Series Analysis, 34, 552–561.
  • Uhlenbeck, G.E. and Ornstein, L.S. (1930). On the theory of the Brownian motion. Physical Review, 36, 823–841.
  • Valdivieso, L., Schoutens, W. and Tuerlinckx, F. (2009). Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type. Statistical Inference for Stochastic Processes, 12, 1–19.
  • Yu, J. (2004). Empirical characteristic function estimation and its applications. Econometric Reviews, 23, 93–123.
La source de données: Dialnet